simulation in computer graphics particles of freiburg –computer science department –computer...

Simulation in Computer Graphics

Particles

Matthias Teschner

Computer Science DepartmentUniversity of Freiburg

University of Freiburg – Computer Science Department – Computer Graphics - 2

introduction

particle motion

finite differences

system of first order ODEs

second order ODE

Outline


sets of particles (particle systems) are used to model time-dependent phenomena such as snow, fire, smoke

Motivation


particles are characterized by mass, position and velocity

forces determine the dynamic behavior

inter-particle forces are neglected

particles can carry arbitrary attributes for rendering purposes, e.g., shape, color, transparency, life time

Motivation

Kolb, Latta, Rezk-Salama


Demo

750,000 particles in XNA, http://www.youtube.com/watch?v=CyAZ2Y7nOTw


introduction

particle motion

finite differences


second order ODE

Outline


quantities relevant for the motion of a particle:

mass

position

velocity

force acting on the particle

force generally depends on position and velocity

Particle Quantities


quantities are considered at discrete time points

particle simulations are concerned with the computation

of unknown future particle quantities

using known current information

Particle Motion


Newton’s Second Law, Newton's motion equation, motion equation of a particle

the force acting on an object is equal to the rate of change of its momentum

constant mass

Governing Equation


is an ordinary differential equation ODE

describes the behavior of in terms of its derivatives with respect to time

numerical integration can be employed to numerically solve the ODE, i.e. to approximate the unknown function

Governing Equation


initial value problem of second order

second-order ODEs can be rewritten as a system of two coupled equations of first order

initial value problem of first order

Governing Equation


functions represent the particle motion

initial values are given

first-order differential equations are given

the functions and their first derivatives are known at

how to compute

Initial Value Problem of First Order


generally depend on positions and velocities

friction / fluid viscosity depend on velocities

spring forces, shear, stretch depend on positions

contact handling forces depend on positions and velocities

can be arbitrarily expensive to compute

consider one particle (particle system) or sets of particles (deformables, fluids)

require additional effort, e.g., contact handling forces detect collisions of particles with obstacles

compute penalty force from penetration depth

Forces


introduction

particle motion

finite differences


second order ODE

Outline


Finite Differences

Taylor-series approximation

continuous ODEs are replaced with discrete finite-difference equations FDEs

O(h2) – truncation or discretization error

O(h) – error order of, e.g., a schemethat employs such approximation


polynomial fitting (line fitting in case of one sample)

which results in

Finite Differences


introduction

particle motion

finite differences


explicit approaches

predictor corrector approaches

implicit approaches

second order ODE

Outline


initialize

numerical integration of position and velocity

Euler Method


Euler step from to

Euler step from to

the position update depends on velocity

the velocity update depends on position and velocity

Coupled Equations


discretization error is defined as the difference between the solution of the ODE and the solution of the FDE

the FDE is consistent, if the discretization error vanishes if the time step h approaches zero

the FDE is stable, if previously introduced errors (discretization, round-off) do not grow within a simulation step

the FDE is convergent, if the solution of the FDE approaches the solution of the ODE

Accuracy and Stability


although the discretization error is diminished by smaller time steps in consistent schemes, the discretization error is introduced in each step of the FD scheme

if previously introduced discretization errors are not amplified by the FD scheme, then it is stable

consistent and stable schemes are convergent

Accuracy and Stability


if stability is influenced by the time step, the FD scheme is conditionally stable

if the FD scheme is stable or unstable for arbitrary time steps, it is unconditionally stable or unstable

ODE, FDE and the parameters influencethe stability of a system

schemes with improved stability work with larger time steps

Stability


larger time steps typically speed up a simulation

smaller time steps can improve the stability

arbitrarily small time steps are not feasible due to round-off errors

for larger time steps, the error is dominated by the discretization error

for smaller time steps, the error is dominated by round-off errors

performance of an FD scheme is trade-off between error order in terms of the time step and computing complexity

Time Step


Second-Order Runge-KuttaMidpoint Method

Euler

• compute the derivative at t0

• approximate f (t0 +h)using the derivative at t0

error

Midpoint Method

• compute the derivative at t0

• compute f(t0 +h/2)

• compute the derivative at t0 +h/2

• approximate f (t0 +h)using the derivative at t0 +h/2

error


compute x’(t)

compute v’(t)

compute x’(t+h/2)

compute v’(t+h/2)with x(t+h/2) and v(t+h/2)

compute x(t+h) with x’(t+h/2)

compute v(t+h) with v’(t+h/2)

Second-Order Runge-Kutta Midpoint Method


compute x’(t)

compute v’(t)

compute x’(t+h)

compute v’(t+h)

c. x(t+h) with x’(t) and x’(t+h)

c. v(t+h) with v’(t) and v’(t+h)

Second-Order Runge-KuttaHeun


compute x’(t)

compute v’(t)

compute x’(t+3h/4)

compute v’(t+3h/4)with x(t+3h/4) and v(t+3h/4)

c. x(t+h) with x’(t) and x’(t+3h/4)

c. v(t+h) with v’(t) and v’(t+3h/4)

Second-Order Runge-Kutta Ralston Method


compute f’(t0) (1) compute f(t0+h/2)

with f (t0) and f’(t0) compute f’(t0+h/2) (2) compute f(t0+h/2)

with f (t0) and f’(t0+h/2) compute f’(t0+h/2) (3) compute f(t0+h)

with f (t0) and f’(t0+h/2) compute f’(t0+h) (4) compute f(t0+h) with f (t0) and a

weighted average of all derivatives (1) – (4)

Fourth-Order Runge-KuttaRunge

error


four derivative computations per time step

error

Fourth-Order Runge-KuttaRunge


four derivative computations per time step

error

Fourth-Order Runge-KuttaKutta


similar performance if the time step for RK 2 is twice the time step for Euler

Does RK allow for faster simulations than Euler?

Performance

Euler Runge-Kutta

• one computation of thederivative per time step

• error

• two (four) computations of the derivative per time step

• error

• allows larger time steps


Euler

one function evaluation

force computation

position and velocity update

Runge-Kutta

multiple function evaluations

computation of auxiliary forces, positions, velocities once for second order

three times for fourth order

position and velocity update

Implementation


introduction

particle motion

finite differences


explicit approaches


implicit approaches

second order ODE

Outline


predict a value from current (and previous) derivatives correct the predicted value with its derivative second-order Adams-Bashforth predictor

second-order Adams-Moulton corrector

based on Lagrange polynomials or Taylor approximations can be efficiently implemented with

two derivative computations per simulation step requires values at previous time steps,

not self-starting

Predictor-Corrector Methods


Adams-Bashforth Predictors


Adams-Moulton Correctors


introduction

particle motion

finite differences


explicit approaches


implicit approaches

second order ODE

Outline


explicit Euler implicit Euler

- one unknown per equation - system of algebraic equations- direct calculation of x(t+h) with many unknowns

and v(t+h) - simultaneous computation of- non-linear equations have x(t+h) and v(t+h)

no effect on the approach - solution of a system of equations- can handle non-analytical, - non-linear equations are commonly

procedural forces linearized to get a system of linear equations

Explicit and Implicit Integration


one-dimensional scalar field

multi-dimensional scalar field

gradient

nabla, del

Linearization of Scalar Functions


multi-dimensional vector field

Jacobi matrix

Linearization of Vector Fields


general form

explicit Euler

implicit Euler

Crank Nicolson

Implicit IntegrationTheta Scheme


rewriting the problem for

force linearization

solving a linear system for

Theta SchemeExample Implementation

In this example,force F depends on x, not on v.


linear system

gradient of a function

with

iterative solution for with initial value

Theta SchemeConjugate Gradient


v0 = v(t) direction d residual r step size v(t+h) = vi

A is symmetric, positive-definite -r0, -d0 gradient of function f di, dj are conjugate,

i.e. diT A dj = 0

ri = 0 in maximal n steps, if v has n components

Conjugate Gradient Method


Euler-Cromer (semi-implicit)

Euler-Cromer

...

computeForces(); //F(t)

velocityEuler(h); //v=v(t+h)=v(t)+ha(t)

positionEuler(h); //x=x(t+h)=x(t)+hv(t+h)

...

Euler

...


positionEuler(h); //x=x(t+h)=x(t)+hv(t)


...


error

can generally handle larger time steps hcompared to Euler

Leap Frog

Leap Frog

initV() // v(0) = v(0) – (h/2)a(0)

...



positionEuler(h); //x=x(t+h)=x(t)+hv(t+h)

...

Euler

...


positionEuler(h); //x=x(t+h)=x(t)+hv(t)


...


introduction

particle motion

finite differences


second order ODE

Outline


function represents the particle motion

initial values are given

second-order differential equation is given

at time , the function and their derivatives are known

how to compute

Initial Value Problem of Second Order


integration methods for integration methods forfirst-order ODEs second-order ODE

(Newton’s motion equation)

Euler, Heun, Ralston, Verlet, velocity Verlet,

Midpoint method, Beeman, Gear,

4th order Runge-Kutta Euler-Cromer, Leap-Frog

Overview of Integration Schemes


Taylor approximations of and

adding both approximations

Verlet


independent of velocity

one derivative computation per time step

efficient to compute, comparatively accurate

third-order in the position

if required, velocity can be computed, e.g. using

velocity is commonly required for collision response or damping

Verlet


one force (derivative) computation per time step

second-order accuracy in position and velocity

equivalent to

Velocity Verlet

Ft+h is computed using xt+h, vt+h


one force (derivative) computation per time step

efficient to compute

third-order accuracy in position and velocity

is computed using

Beeman


position velocity acceleration

Gear Integration


Gear Integration


Gear - Prediction


error

error correction coefficients

Gear - Correction

k = 0 k = 1 k = 2 k = 3 k = 4 k = 5

20

3

360

2511

18

11

6

1

60

1


initialization

integration

prediction

error estimation

correction

Gear - Implementation


method force comp. error order error order

per time step position velocity

Euler 1 1 1

RK 2nd order 2 2 2

RK 4th order 4 4 4

Verlet 1 3 1

Velocity Verlet 1 2 2

Beeman 1 3 3

Comparison – Explicit Schemes


Comparison – Explicit Schemes

methods for first-order ODEs accuracy corresponds to computing complexity position and velocity have the same error order

methods for second-order ODEs improved accuracy with minimal computing complexity error order might differ for position and velocity

implicit methods cannot be compared this way do not only compute forces (derivatives) commonly require to solve a linear system improved stability even for low error orders,

implicit Euler with error order one can be unconditionally stable, e. g. for harmonic oscillators (springs)


Advantages / Disadvantages

explicit methods simple to set up and program fast computation per integration step suitable for parallel architectures small time steps required for stability many computing steps required for a given time interval t

implicit methods stability is maintained for large time steps require less steps for a given interval t large time steps can cause large truncation errors complicate to set up less flexible, problems with non-analytical forces large computing time per integration step


Advantages / Disadvantages

predictor-corrector methods not self-starting have to be re-initialized in case of discontinuities,

e. g. due to collision response

in general, implicit methods are more robust (stable) compared to explicit methods if an explicit scheme is not conditionally or unconditionally

stable it cannot be used regardless of its efficiency

explicit methods can be computed efficiently which is essential if frequent updates are required if an implicit scheme cannot be computed at

interactive rates, it cannot be used in interactive applications regardless of the time step


Summary

motion equation for a mass point second-order differential equation coupled system of first-order differential equations

numerical integration initial values xt and vt

approximate integration of v and x through time with time step h

integration schemes Euler, Runge-Kutta 2nd , Runge-Kutta 4th

Crank-Nicolson, implicit Euler, Euler-Cromer, Leap-Frog Gear, Verlet, velocity Verlet, Beeman

trade-off between accuracy and computing cost goal: maximizing the ratio of time step and computing cost

simulation in computer graphics particles of freiburg –computer science department –computer...

Documents